nLab Sandbox

positions of horizontal steps positions of vertical steps μ 1<μ 2<<μ p ν 1<ν 2<<ν p \begin{array}{cc} \text{positions of horizontal steps} & \text{positions of vertical steps} \\ \mu_1 \lt \mu_2 \lt \cdots \lt \mu_p & \nu_1 \lt \nu_2 \lt \cdots \lt \nu_p \end{array}

Definition

(ScholzeWeinstein20, Definition 6.2.1)

Let RR be a perfectoid ring. The tilt of RR is

R =limxx pRR^{\flat}=\underset {\underset{x\mapsto x^{p}}{\leftarrow}} {\lim}R

A priori this is a topological multiplicative monoid, so turn it into a topological ring we equip it with the addition structure given by

(x+y) (i)=lim n(x (i+n)+y (i+n)) p n.(x+y)^{(i)}=\lim_{n\to\infty}(x^{(i+n)}+y^{(i+n)})^{p^{n}}.

Last revised on December 3, 2022 at 18:02:15. See the history of this page for a list of all contributions to it.